Value Groups and Residue Fields of Models of Real Exponentiation

TL;DR

This paper characterizes which triples of residue fields, value groups, and exponential functions can be realized in models of real exponentiation, especially focusing on countable and saturated cases, and explores residue fields of o-minimal exponential fields.

Contribution

It provides a complete characterization of realizable triples in models of real exponentiation for specific types of value groups and shows residue fields of certain o-minimal exponential fields are models of real exponentiation.

Findings

Characterization for countable G cases.

Characterization for uncountably saturated G cases.

Residue fields of o-minimal exponential fields satisfy real exponentiation.

Abstract

Let $F$ be an archimedean field, $G$ a divisible ordered abelian group and $h$ a group exponential on $G$. A triple $(F,G,h)$ is realised in a non-archimedean exponential field $(K,\exp)$ if the residue field of $K$ under the natural valuation is $F$ and the induced exponential group of $(K,\exp)$ is $(G,h)$. We give a full characterisation of all triples $(F,G,h)$ which can be realised in a model of real exponentiation in the following two cases: i) $G$ is countable. ii) $G$ is of cardinality $\kappa$ and $\kappa$-saturated for an uncountable regular cardinal $\kappa$ with $\kappa^{<\kappa} = \kappa$. Moreover, we show that for any o-minimal exponential field $(K, \exp)$ satisfying the differential equation $\exp' = \exp$, its residue exponential field is a model of real exponentiation.